I'm personally lukewarm about this question, but that's just me. About your "meta question", for future reference, please ask them on Meta.mathoverflow (preferably before you ask questions here that you are not sure is appropriate). For this question (and other etymological ones), I've opened a meta thread: tea.mathoverflow.net/discussion/750/etymological-questions please discuss over there.
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Willie WongNov 7 '10 at 11:59

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Great question with interesting answers. What are the others?
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RomeoNov 7 '10 at 16:54

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Most etymologies are not that hard to track down, so unless the answers contain some insightful mathematical discussion spun off the strict etymologies, I doubt they would add much value to MO.
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Thierry ZellNov 7 '10 at 17:12

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Yuji- I would recommend against creating a large number of such questions. I would also recommend that you check obvious sources first, and only come to MO when you have been unable to find the information elsewhere. For example, this question is basically unsuitable on the grounds that it is quite adequately answered by Wikipedia.
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Ben Webster♦Nov 7 '10 at 18:02

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The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic." Dickson calls the group the "Abelian linear group" in homage to Abel who first studied it.

The following is from page 1 of
Lectures on symplectic geometry, by Ana Cannas da Silva:

As a curiousity, note that two centuries ago the name symplectic did not exist.
If you consult a major English dictionary, you are likely to find that
symplectic is the name of a bone in a fish's head. However ... the word
symplectic in mathematics was coined by Weyl who substituted the Latin root incomplex by the corresponding Greek root, in order to label the symplectic group.

Interesting remark. And she wrote, very rightly, "the word symplectic in mathematics was coined by Weyl", as the adjective συμπλεκτικός did exist in classic Greek. Since Weyl received a humanistic education, no doubt he was quite reluctant to create artificially a new word.
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Pietro MajerNov 7 '10 at 21:16

There is a brief explanation here. It looks like the term was coined by Weyl, and was a result of modifying the Greek root "comp" from "complex" to the equivalent Latin root "symp". This is a pretty obscure way to coin a word if you ask me!

-1 for saying that "comp" (or "symp") is a Latin root. "Complexus" is originally the p.p. of the verb complector (to embrace, thus, to put together into a whole etc), which is a compound of cum (with) and plecto, and that exactly corresponds to the Greek verb συμπλέκω, compound of σύν and πλέκω. Indeed Weyl did not "coined" the (already existing) Greek term, but only gave it a new mathematical meaning, in analogy to the (modern) mathematical meaning of the Latin term. I'm not sure if technically this can be called a calque.
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Pietro MajerNov 7 '10 at 15:20

The word "sum-plectic" as a greek translation of "com-plexus" was needed also to differentiate the study of "complex geometry" (complex numbers etc) from the study of "complexes de droites" (the geometry of 'line complexes') where the Plücker coordinates are associated to a natural symplectic structure on the space of affines lines in any euclidean space.

The concept of "differential symplectic geometry" has been introduced I believe by J.-M. Souriau in is 1953 paper

From Pietro Majer's comments I learn that "symplectic" is the past participle of a classic Greek verb which means "to embrace".
Consequently I would just remark then how surprising is the effectiveness of this adjective to reflect the pervasiveness of the ideas from symplectic geometry in modern mathematics, which interconnect many different subjects.
Infact, from the introduction to the paper "Symplectic Geometry" by A.Weinstein, I quote:

I think it is not unfair to say that symplectic geometry is of interest today, not so much as a theory in itself, but rather because of a series of remarkable "transforms" which connect it with various areas of analysis.